Now we use the main theorem of [1]} to find a subspace of \(S_1(\Gamma _0(N), \mathbb {F}_2)^{\operatorname{Katz}}\)  that maps under \(V\)  to \(S_2(\Gamma _0(N), \mathbb {F}_2)_{\mathfrak {m}}\cap \operatorname{Ker}A\theta \) . As in Section REF , we have \({\operatorname{an}}\) -equivariance, and so the maximal ideal \(\mathfrak {m}\)  has an exact analogue in \({1,\operatorname{an}}\)  and we land in the subspace \(S_1(\Gamma _0(N), \mathbb {F}_2)_{\mathfrak {m}}^{\operatorname{Katz}}\) . We may not obtain the whole subspace because, while \(Vf\)  is in the kernel of \(A\theta \)  for all \(f\in S_1(\Gamma _0(N), \mathbb {F}_2)_{\mathfrak {m}}^{\operatorname{Katz}}\) , we don't know that it's a form that is the reduction of a \(\mathbb {Z}_2\)  form, which is what \({\operatorname{an}}_{\mathfrak {m}}\)  parametrizes. In this section we will prove that the space of Katz forms of weight 2 actually are all standard forms.
